# Section 4.1 Calculus 2

Planar Parametrizations

## 4.1 Planar Parametrizations

### 4.1.1 Parametric Equations

• If $$x(t),y(t)$$ are both defined as functions of $$t$$ over some interval $$I$$ of real numbers, then the set of points $$\{(x(t),y(t)):t\in I\}$$ is the parametric curve with a system of parametric equations $$x(t),y(t)$$. If $$I$$ is omitted, then it is assumed that $$t$$ belongs to subset of the interval $$(-\infty,\infty)$$ where both $$x(t),y(t)$$ are well-defined.
• Example Plot the parametric curve $$x=\cos t,y=\sin t$$ for $$0\leq t\leq 2\pi$$, first by using a chart of $$t,x,y$$ values, then by expressing the curve as an equation of $$x,y$$.
• Example Show that the systems of parametric equations $$x_0=t,y_0=t^2$$ and $$x_1=2t-2,y_1=4t^2-8t+4$$ share the same parametric curve.

### 4.1.2 Parametrizing Curves Defined by Functions

• The curve $$y=f(x)$$ where $$x$$ belongs to the interval $$I$$ may be easily parametrized left-to-right by the system of parametric equations $$x=t,y=f(t)$$ where $$t$$ also belongs to $$I$$.
• Example Give a system of parametric equations for the curve $$y=\ln x$$ from $$(1,0)$$ to $$(e^2,2)$$.

### 4.1.3 Parametrizing Line Segments and Circles

• The line segment joining the points $$(x_0,y_0),(x_1,y_1)$$ may be parametrized by $$x=x_0+(x_1-x_0)t,y=y_0+(y_1-y_0)t$$ where $$0\leq t\leq 1$$.
• Example Give a system of parametric equations for the line segment joining $$(2,-3)$$ and $$(-1,4)$$.
• Example Give two different systems of parametric equations for the portion of the line $$y=3x-2$$ between $$x=-1$$ and $$x=2$$.
• The full line may be obtained with the same equations by allowing $$t$$ to range over all real numbers.
• The circle with center $$(x_0,y_0)$$ and radius $$r$$ may be parametrized counter-clockwise by $$x=x_0+r\cos\theta,y=y_0+r\sin\theta$$ where $$0\leq\theta\leq2\pi$$.
• Example Give a system of parametric equations for the circle $$x^2+y^2=9$$.
• The circle with center $$(x_0,y_0)$$ and radius $$r$$ may be parametrized clockwise by $$x=x_0+r\sin\theta,y=y_0+r\cos\theta$$ where $$0\leq\theta\leq2\pi$$.
• Example Give a system of parametric equations for the circle $$(x-3)^2+(y+4)^2=25$$ beginning at $$(3,1)$$ and moving clockwise.

### Exercises for 4.1

1. Plot the parametric curve $$x=2-t^2,y=2t^2$$ for $$0\leq t\leq 3$$, first by using a chart of $$t,x,y$$ values, then by expressing the curve as an equation of $$x,y$$.
2. Plot the parametric curve $$x=3^t,y=3^{-t}$$ for $$-\infty<t<\infty$$, first by using a chart of $$t,x,y$$ values, then by expressing the curve as an equation of $$x,y$$.
3. Show that the systems of parametric equations $$x_0=t+2,y_0=e^2e^t$$ and $$x_1=\ln t,y_1=t$$ share the same parametric curve. Then plot that curve.
4. Give a system of parametric equations for the curve $$y=\cosh x$$ from $$(-\ln 2,5/4)$$ to $$(\ln 2,5/4)$$.
5. Give a system of parametric equations for the line segment joining $$(0,-4)$$ and $$(3,5)$$.
6. Give a system of parametric equations for the line segment joining $$(1,2)$$ and $$(-3,3)$$.
7. Give two different systems of parametric equations for the portion of the line $$y=4-3x$$ between $$x=-2$$ and $$x=3$$.
8. Let $$a<b$$. Find a system of parametric equations which parametrizes the planar curve $$y=f(x)$$ right-to-left from $$x=b$$ to $$x=a$$.
9. Give a system of parametric equations that parametrize the circle $$x^2+(y+1)^2=9$$ counter-clockwise.
10. Give a system of parametric equations that parametrize the arc of the circle $$(x-2)^2+(y+3)^2=4$$ clockwise from $$(2,-5)$$ to $$(4,-3)$$.

Solutions

#### Textbook References

• University Calculus: Early Transcendentals (3rd Ed)
• 10.1